Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,26,31,32)(27,35,36,33)(28,39,40,29)(30,37,38,34);(2,6)(3,7)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(26,32)(27,36)(29,39)(34,37);(1,3,5,7)(2,4,6,8)(9,11,14,16)(10,15,18,12)(13,21,22,19)(17,20,24,23)(25,27,31,36)(26,33,32,35)(28,37,40,34)(29,38,39,30);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,24)(21,22)(25,28)(26,29)(27,37)(30,35)(31,40)(32,39)(33,38)(34,36);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,29)(26,28)(27,38)(30,36)(31,39)(32,40)(33,37)(34,35);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,30,31,38)(26,34,32,37)(27,29,36,39)(28,35,40,33)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839344);